A satellite A has twice the mass of satellite B. The satellites orbit a planet with the same orbital radius. The orbital velocity of the satellite A compared to B is:

Question

BlackTom AI · Accepted Answer

For a circular orbit the equilibrium condition is that the gravitational force provides the required centripetal force. Using that:

1) Relevant concepts
- Newton’s law of universal gravitation: F_grav = G M m / r^2
- Centripetal force for circular motion: F_cent = m v^2 / r
- Important point: m is the satellite’s mass, M is the planet’s mass, r is orbital radius. For a given planet and radius, M and r are fixed.

2) Step-by-step reasoning and algebra
Set gravitational force equal to centripetal force:
G M m / r^2 = m v^2 / r

3) Intermediate algebraic steps
- Cancel the satellite mass m from both sides (it appears multiplicatively):
G M / r^2 = v^2 / r
- Multiply both sides by r:
G M / r = v^2
- Take square root:
v = sqrt(G M / r)

4) Why this gives the correct answer
The expression v = sqrt(GM/r) contains no factor of the satellite mass m. Therefore the orbital speed depends only on the planet’s mass M and the orbital radius r, not on the satellite’s mass. If satellite A has twice the mass of B but both orbit at the same radius around the same planet, their speeds are identical.

5) Why the other choices are incorrect
- a (Four times greater): would require v ∝ m or v ∝ sqrt(m)^2; no such dependence exists.
- c (Half as much): implies v ∝ 1/m, which is false.
- d (1.4 times greater): 1.4 ≈ sqrt(2); this would be relevant if some quantity scaled as sqrt(m) but speed does not depend on m.
- e (Twice greater): implies v ∝ m linearly, which is incorrect.

Conclusion: Because the satellite mass cancels out in the orbital speed formula v = sqrt(GM/r), satellite A and B have the same orbital velocity. This matches the provided answer: b. The same.

A satellite A has twice the mass of satellite B. The satellites orbit a planet with the same orbital radius. The orbital velocity of the satellite A compared to B is:Single choice

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